You may not be able to see the water moving. As long as the engine is not overheating nothing to worry about.
It's very rare to have a water pump fail to pump. They may leak, have bearing failure, but still pump. Only if the impeller falls off will they stop pumping, extremely unlikely.
It's possible the radiator is clogged, but that would be indicated by overheating under heavy load. Again, the air coming through would be cold when the engine was hot.